So may I say: "I wish you on that special day (on your name-day?) many years!"? Is it correct?Or how can I say it another way so the person could understand that I want to congratulate him on this happy day?

I just walk up to the person and say Many Years. Note: On the old calendar today is the synxis of Michael and Gabriel. So you go up to your Orthodox friend named Mike or Gabrielle and you say, "Many Years". Say to a non-Orhtodox friend and they may look at you like you have two heads.

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Save us o' Son of God, who art risen from the dead, as we sing to thee Alleluia!

(In my native Ukrainian, it's "Mnohaya Lita." BTW, note that according to the pre-1917 Russian spelling rules, the "e" in "leta" was literated by a special letter "yat'," signifying that in Old Church Slavonic this sound was not quite "e").

Nektarios, I am not sure if it's "always," but I can say that it's "i" in many cases. The Russian "les" in Ukrainian is, indeed, "lis" (forest), and "belyj" is, indeed, "bilyj" (white); "sneg" is "snih" (snow), "belka" is "bilka" (squirrel), "nezhnyj" is "nizhnyj" (tender), etc.

Why would she want to know that if she asked in the "Practice English" forum?

I think it shows that while English is the international language and influencing many languages, it is also being changed by its contact with other languages. And this is especially true of Orthodox vocabulary and customary greetings since it is so new to the anglophone world. For my recently celebrated name day I received wishes of "Happy name day", "Many years" "Chronia Polla" and "mnogaya leta" - oddly enough the last two tended to come from native English speakers and the former two from non-native English speakers. In that sense I think Pan Heorhij answered the question best by demonstrating that in Anglophone culture the concept of name days is very underdeveloped and is still borrowing rather than generating a "native" response.